1 Motivation -- 1–1 Introduction -- 1–2 Miscellaneous Examples -- 1–3 Conclusion: Our Present Aim -- 2 The Initial Axioms -- 2–1 Some Logical Geography -- 2–2 Commutative Bands -- 2–3 Isotone Functions -- 2–4 Belts -- 2–5 Belt Homomorphisms -- 2–6 Types of Belt -- 2–7 Dual Addition -- 2–8 Duality for Belts -- 2–9 Some Specific Cases -- 3 Opening and Closing -- 3–1 The Operations ??,???,??,??? -- 3–2 The Principle of Closing -- 3–3 The Principle of Opening -- 4 The Principal Interpretation -- 4–1 Blogs -- 4–2 The Principal Interpretation -- 4–3 The 3-element Blog ? -- 4–4 Further Properties of Blogs -- 5 The Spaces En and ?mn -- 5–1 Band-Spaces -- 5–2 Two-Sided Spaces -- 5–3 Function Spaces -- 5–4 Matrix Algebra -- 5–5 The Identity Matrix -- 5–6 Matrix Transformations -- 5–7 Further Notions -- 6 Duality for Matrices -- 6–1 The Dual Operations -- 6–2 Some Matrix Inequalities -- 7 Conjugacy -- 7–1 Conjugacy for Belts -- 7–2 Conjugacy for Matrices -- 7–3 Two Examples -- 8 AA* Relations -- 8–1 Pre-residuation -- 8–2 Alternating AA* Products -- 8–3 Modified AA* Products -- 8–4 Some Bijections -- 8–5 A Worked Example -- 9 Some Schedule Algebra -- 9–1 Feasibility and Compatibility -- 9–2 The Float -- 9–3 A Worked Example -- 10 Residuation and Representation -- 10–1 Some Residuation Theory -- 10–2 Residuomorphisms -- 10–3 Representation Theorems -- 10–4 Representation for Matrices -- 10–5 Analogy with Hilbert Space -- 11 Trisections -- 11–1 The Demands of Reality -- 11–2 Trisections -- 11–3 Convex Subgroups -- 11–4 The Linear Case -- 11–5 Two Examples -- 12 ? ø — Astic Matrices -- 12–1 ?ø — Asticity -- 12–2 The Generalised Question 2 -- 13 / — Existence -- 13–1 / — Existence Defined -- 13–2 Compatible Trisections -- 13–3 Dually ? ø — astic Matrices -- 13–4 / — Defined Residuomorphisms -- 13–5 Omn As Operators -- 13–6 Some Questions Answered -- 14 The Equation A ? x = b Over a Bldg -- 14–1 Some Preliminaries -- 14–2 The Principal Solution -- 14–3 The Boolean Case -- 15 Linear Equations over a Linear Bldg -- 15–1 All Solutions of (14–3) -- 15–2 Proving the Procedure -- 15–3 Existence and Uniqueness -- 15–4 A Linear Programming Criterion -- 15–5 Left-Right Variants -- 16 Linear Dependence -- 16–1 Linear Dependence Over El -- 16–2 The A Test -- 16–3 Some Dimensional Anomalies -- 16–4 Strong Linear Independence -- 17 Rank of Matrices -- 17–1 Regular Matrices -- 17–2 Matrix Rank Over A Linear Blog -- 17–3 Existence of Rank -- 18 Seminorms on En -- 18–1 Column-Spaces -- 18–2 Seminorms -- 18–3 Spaces of Bounded Seminorm -- 19 Some Matrix Spaces -- 19–1 Matrix Seminorms -- 19–2 Matrix Spaces -- 19–3 The Role of Conjugacy -- 20 The Zero-Lateness Problem -- 20–1 The Principal Solution -- 20–2 Case of Equality -- 20–3 Critical Paths -- 21 Projections -- 21–1 Congruence Classes -- 21–2 Operations in Rang A -- 21–3 Projection Matrices -- 22 Definite and Metric Matrices -- 22–1 Some Graph Theory -- 22–2 Definite Matrices -- 22–3 Metric Matrices -- 22–4 The Shortest Distance Matrix -- 23 Fundamental Eigenvectors -- 23–1 The Eigenproblem -- 23–2 Blocked Matrices -- 23–3 ø-Astic Definite Matrices -- 24 Aspects of the Eigenproblem -- 24–1 The Eigenspace -- 24–2 Directly Similar Matrices -- 24–3 Structure of the Eigenspace -- 25 Solving the Eigenproblem -- 25–1 The Parameter ?(A) -- 25–2 Properties of ?(A) -- 25–3 Necessary and Sufficient Conditions -- 25–4 The Computational Task -- 25–5 An Extended Example -- 26 Spectral Inequalities -- 26–1 Preliminary Inequalities -- 26–2 Spectral Inequality -- 26–3 The Other Eigenproblems -- 26–4 More Spectral Inequalities -- 26–5 The Principal Interpretation -- 27 The Orbit -- 27–1 Increasing Matrices -- 27–2 The Orbit -- 27–3 The Orbital Matrix -- 27–4 A Practical Case -- 27–5 More General Situations -- 27–6 Permanents -- 28 Standard Matrices -- 28–1 Direct Similarity -- 28–2 Invertible Matrices -- 28–3 Equivalence of Matrices -- 28–4 Equivalence and Rank -- 28–5 Rank of ? -- 29 References and Notations -- 29–1 Previous Publications -- 29–2 Related References -- 29–3 List of Notations -- 29–4 List of Definitions.
